LMFDB - Embedded newform 504.6.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$80.8334451857$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{114}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 114 $ x^2 - 114
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.6771$ of defining polynomial
Character	$\chi$	$=$	504.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-78.0625 q^{5} -49.0000 q^{7} +746.437 q^{11} +9.75012 q^{13} -1754.69 q^{17} -603.250 q^{19} +3175.31 q^{23} +2968.75 q^{25} -3227.50 q^{29} +7881.75 q^{31} +3825.06 q^{35} +12362.7 q^{37} -3699.56 q^{41} +16915.0 q^{43} -658.130 q^{47} +2401.00 q^{49} -27891.1 q^{53} -58268.7 q^{55} -7456.38 q^{59} -43705.7 q^{61} -761.119 q^{65} -28643.5 q^{67} +9244.93 q^{71} -29816.5 q^{73} -36575.4 q^{77} +32928.5 q^{79} -40524.5 q^{83} +136975. q^{85} -41351.3 q^{89} -477.756 q^{91} +47091.2 q^{95} -110780. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 28 q^{5} - 98 q^{7} + 596 q^{11} + 532 q^{13} - 2100 q^{17} - 2744 q^{19} + 3660 q^{23} + 2350 q^{25} + 720 q^{29} + 3976 q^{31} + 1372 q^{35} + 6788 q^{37} + 4004 q^{41} + 19480 q^{43} - 21560 q^{47}+ \cdots - 36036 q^{97}+O(q^{100}) $ 2 * q - 28 * q^5 - 98 * q^7 + 596 * q^11 + 532 * q^13 - 2100 * q^17 - 2744 * q^19 + 3660 * q^23 + 2350 * q^25 + 720 * q^29 + 3976 * q^31 + 1372 * q^35 + 6788 * q^37 + 4004 * q^41 + 19480 * q^43 - 21560 * q^47 + 4802 * q^49 - 57576 * q^53 - 65800 * q^55 - 22344 * q^59 - 38724 * q^61 + 25384 * q^65 + 7288 * q^67 - 3932 * q^71 + 19292 * q^73 - 29204 * q^77 + 15632 * q^79 - 22624 * q^83 + 119688 * q^85 - 112812 * q^89 - 26068 * q^91 - 60080 * q^95 - 36036 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−78.0625	−1.39642	−0.698212	−	0.715891i	$-0.746021\pi$
$5$	−78.0625	−1.39642	−0.698212	+	0.715891i	$0.746021\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	746.437	1.85999	0.929997	−	0.367567i	$-0.119809\pi$
$11$	746.437	1.85999	0.929997	+	0.367567i	$0.119809\pi$
$12$	0	0
$13$	9.75012	0.0160012	0.00800058	−	0.999968i	$-0.497453\pi$
$13$	9.75012	0.0160012	0.00800058	+	0.999968i	$0.497453\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1754.69	−1.47257	−0.736287	−	0.676669i	$-0.763423\pi$
$17$	−1754.69	−1.47257	−0.736287	+	0.676669i	$0.763423\pi$
$18$	0	0
$19$	−603.250	−0.383366	−0.191683	−	0.981457i	$-0.561395\pi$
$19$	−603.250	−0.383366	−0.191683	+	0.981457i	$0.561395\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3175.31	1.25160	0.625802	−	0.779982i	$-0.284772\pi$
$23$	3175.31	1.25160	0.625802	+	0.779982i	$0.284772\pi$
$24$	0	0
$25$	2968.75	0.950000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3227.50	−0.712641	−0.356321	−	0.934364i	$-0.615969\pi$
$29$	−3227.50	−0.712641	−0.356321	+	0.934364i	$0.615969\pi$
$30$	0	0
$31$	7881.75	1.47305	0.736526	−	0.676409i	$-0.236465\pi$
$31$	7881.75	1.47305	0.736526	+	0.676409i	$0.236465\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3825.06	0.527799
$36$	0	0
$37$	12362.7	1.48460	0.742302	−	0.670065i	$-0.233734\pi$
$37$	12362.7	1.48460	0.742302	+	0.670065i	$0.233734\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3699.56	−0.343709	−0.171854	−	0.985122i	$-0.554976\pi$
$41$	−3699.56	−0.343709	−0.171854	+	0.985122i	$0.554976\pi$
$42$	0	0
$43$	16915.0	1.39509	0.697543	−	0.716543i	$-0.254277\pi$
$43$	16915.0	1.39509	0.697543	+	0.716543i	$0.254277\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−658.130	−0.0434577	−0.0217289	−	0.999764i	$-0.506917\pi$
$47$	−658.130	−0.0434577	−0.0217289	+	0.999764i	$0.506917\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−27891.1	−1.36388	−0.681940	−	0.731408i	$-0.738864\pi$
$53$	−27891.1	−1.36388	−0.681940	+	0.731408i	$0.738864\pi$
$54$	0	0
$55$	−58268.7	−2.59734
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7456.38	−0.278867	−0.139434	−	0.990231i	$-0.544528\pi$
$59$	−7456.38	−0.278867	−0.139434	+	0.990231i	$0.544528\pi$
$60$	0	0
$61$	−43705.7	−1.50388	−0.751941	−	0.659230i	$-0.770882\pi$
$61$	−43705.7	−1.50388	−0.751941	+	0.659230i	$0.770882\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−761.119	−0.0223444
$66$	0	0
$67$	−28643.5	−0.779541	−0.389770	−	0.920912i	$-0.627446\pi$
$67$	−28643.5	−0.779541	−0.389770	+	0.920912i	$0.627446\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9244.93	0.217650	0.108825	−	0.994061i	$-0.465291\pi$
$71$	9244.93	0.217650	0.108825	+	0.994061i	$0.465291\pi$
$72$	0	0
$73$	−29816.5	−0.654861	−0.327431	−	0.944875i	$-0.606183\pi$
$73$	−29816.5	−0.654861	−0.327431	+	0.944875i	$0.606183\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−36575.4	−0.703012
$78$	0	0
$79$	32928.5	0.593614	0.296807	−	0.954938i	$-0.404078\pi$
$79$	32928.5	0.593614	0.296807	+	0.954938i	$0.404078\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−40524.5	−0.645687	−0.322844	−	0.946452i	$-0.604639\pi$
$83$	−40524.5	−0.645687	−0.322844	+	0.946452i	$0.604639\pi$
$84$	0	0
$85$	136975.	2.05634
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−41351.3	−0.553368	−0.276684	−	0.960961i	$-0.589236\pi$
$89$	−41351.3	−0.553368	−0.276684	+	0.960961i	$0.589236\pi$
$90$	0	0
$91$	−477.756	−0.00604787
$92$	0	0
$93$	0	0
$94$	0	0
$95$	47091.2	0.535341
$96$	0	0
$97$	−110780.	−1.19546	−0.597728	−	0.801699i	$-0.703930\pi$
$97$	−110780.	−1.19546	−0.597728	+	0.801699i	$0.703930\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−80973.4	−0.789840	−0.394920	−	0.918716i	$-0.629228\pi$
$101$	−80973.4	−0.789840	−0.394920	+	0.918716i	$0.629228\pi$
$102$	0	0
$103$	104902.	0.974298	0.487149	−	0.873319i	$-0.338037\pi$
$103$	104902.	0.974298	0.487149	+	0.873319i	$0.338037\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−193878.	−1.63708	−0.818540	−	0.574450i	$-0.805216\pi$
$107$	−193878.	−1.63708	−0.818540	+	0.574450i	$0.805216\pi$
$108$	0	0
$109$	−154948.	−1.24917	−0.624584	−	0.780957i	$-0.714732\pi$
$109$	−154948.	−1.24917	−0.624584	+	0.780957i	$0.714732\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−91578.5	−0.674679	−0.337340	−	0.941383i	$-0.609527\pi$
$113$	−91578.5	−0.674679	−0.337340	+	0.941383i	$0.609527\pi$
$114$	0	0
$115$	−247873.	−1.74777
$116$	0	0
$117$	0	0
$118$	0	0
$119$	85979.7	0.556581
$120$	0	0
$121$	396118.	2.45958
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12197.3	0.0698216
$126$	0	0
$127$	110234.	0.606465	0.303233	−	0.952917i	$-0.401934\pi$
$127$	110234.	0.606465	0.303233	+	0.952917i	$0.401934\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−361365.	−1.83979	−0.919895	−	0.392166i	$-0.871726\pi$
$131$	−361365.	−1.83979	−0.919895	+	0.392166i	$0.871726\pi$
$132$	0	0
$133$	29559.3	0.144899
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−293644.	−1.33665	−0.668327	−	0.743867i	$-0.732989\pi$
$137$	−293644.	−1.33665	−0.668327	+	0.743867i	$0.732989\pi$
$138$	0	0
$139$	92541.4	0.406255	0.203128	−	0.979152i	$-0.434889\pi$
$139$	92541.4	0.406255	0.203128	+	0.979152i	$0.434889\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7277.85	0.0297621
$144$	0	0
$145$	251946.	0.995149
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13179.0	−0.0486313	−0.0243157	−	0.999704i	$-0.507741\pi$
$149$	−13179.0	−0.0486313	−0.0243157	+	0.999704i	$0.507741\pi$
$150$	0	0
$151$	−304869.	−1.08811	−0.544053	−	0.839051i	$-0.683111\pi$
$151$	−304869.	−1.08811	−0.544053	+	0.839051i	$0.683111\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−615269.	−2.05701
$156$	0	0
$157$	−64050.2	−0.207382	−0.103691	−	0.994610i	$-0.533065\pi$
$157$	−64050.2	−0.207382	−0.103691	+	0.994610i	$0.533065\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−155590.	−0.473062
$162$	0	0
$163$	617219.	1.81958	0.909788	−	0.415072i	$-0.136244\pi$
$163$	617219.	1.81958	0.909788	+	0.415072i	$0.136244\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−494693.	−1.37260	−0.686301	−	0.727318i	$-0.740767\pi$
$167$	−494693.	−1.37260	−0.686301	+	0.727318i	$0.740767\pi$
$168$	0	0
$169$	−371198.	−0.999744
$170$	0	0
$171$	0	0
$172$	0	0
$173$	701459.	1.78191	0.890957	−	0.454088i	$-0.150035\pi$
$173$	701459.	1.78191	0.890957	+	0.454088i	$0.150035\pi$
$174$	0	0
$175$	−145469.	−0.359066
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−197449.	−0.460598	−0.230299	−	0.973120i	$-0.573971\pi$
$179$	−197449.	−0.460598	−0.230299	+	0.973120i	$0.573971\pi$
$180$	0	0
$181$	869311.	1.97233	0.986163	−	0.165777i	$-0.0530133\pi$
$181$	869311.	1.97233	0.986163	+	0.165777i	$0.0530133\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−965066.	−2.07314
$186$	0	0
$187$	−1.30976e6	−2.73898
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−119524.	−0.237068	−0.118534	−	0.992950i	$-0.537819\pi$
$191$	−119524.	−0.237068	−0.118534	+	0.992950i	$0.537819\pi$
$192$	0	0
$193$	170916.	0.330285	0.165143	−	0.986270i	$-0.447192\pi$
$193$	170916.	0.330285	0.165143	+	0.986270i	$0.447192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	715090.	1.31279	0.656394	−	0.754418i	$-0.272081\pi$
$197$	715090.	1.31279	0.656394	+	0.754418i	$0.272081\pi$
$198$	0	0
$199$	−715708.	−1.28116	−0.640580	−	0.767892i	$-0.721306\pi$
$199$	−715708.	−1.28116	−0.640580	+	0.767892i	$0.721306\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	158147.	0.269353
$204$	0	0
$205$	288797.	0.479963
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−450289.	−0.713059
$210$	0	0
$211$	−880061.	−1.36084	−0.680420	−	0.732823i	$-0.738202\pi$
$211$	−880061.	−1.36084	−0.680420	+	0.732823i	$0.738202\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.32043e6	−1.94813
$216$	0	0
$217$	−386206.	−0.556762
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17108.4	−0.0235629
$222$	0	0
$223$	514501.	0.692826	0.346413	−	0.938082i	$-0.387400\pi$
$223$	514501.	0.692826	0.346413	+	0.938082i	$0.387400\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	873242.	1.12479	0.562393	−	0.826870i	$-0.309881\pi$
$227$	873242.	1.12479	0.562393	+	0.826870i	$0.309881\pi$
$228$	0	0
$229$	−1.11239e6	−1.40174	−0.700869	−	0.713290i	$-0.747204\pi$
$229$	−1.11239e6	−1.40174	−0.700869	+	0.713290i	$0.747204\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.21568e6	−1.46700	−0.733499	−	0.679691i	$-0.762114\pi$
$233$	−1.21568e6	−1.46700	−0.733499	+	0.679691i	$0.762114\pi$
$234$	0	0
$235$	51375.2	0.0606854
$236$	0	0
$237$	0	0
$238$	0	0
$239$	462246.	0.523454	0.261727	−	0.965142i	$-0.415708\pi$
$239$	462246.	0.523454	0.261727	+	0.965142i	$0.415708\pi$
$240$	0	0
$241$	−247817.	−0.274845	−0.137423	−	0.990513i	$-0.543882\pi$
$241$	−247817.	−0.274845	−0.137423	+	0.990513i	$0.543882\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−187428.	−0.199489
$246$	0	0
$247$	−5881.76	−0.00613430
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.25628e6	1.25864	0.629322	−	0.777145i	$-0.283333\pi$
$251$	1.25628e6	1.25864	0.629322	+	0.777145i	$0.283333\pi$
$252$	0	0
$253$	2.37017e6	2.32798
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−330237.	−0.311884	−0.155942	−	0.987766i	$-0.549841\pi$
$257$	−330237.	−0.311884	−0.155942	+	0.987766i	$0.549841\pi$
$258$	0	0
$259$	−605775.	−0.561128
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−187729.	−0.167356	−0.0836781	−	0.996493i	$-0.526667\pi$
$263$	−187729.	−0.167356	−0.0836781	+	0.996493i	$0.526667\pi$
$264$	0	0
$265$	2.17725e6	1.90456
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.02334e6	−0.862264	−0.431132	−	0.902289i	$-0.641886\pi$
$269$	−1.02334e6	−0.862264	−0.431132	+	0.902289i	$0.641886\pi$
$270$	0	0
$271$	−649145.	−0.536931	−0.268466	−	0.963289i	$-0.586517\pi$
$271$	−649145.	−0.536931	−0.268466	+	0.963289i	$0.586517\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.21599e6	1.76699
$276$	0	0
$277$	1.40211e6	1.09795	0.548973	−	0.835840i	$-0.315019\pi$
$277$	1.40211e6	1.09795	0.548973	+	0.835840i	$0.315019\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.63255e6	−1.23339	−0.616697	−	0.787201i	$-0.711530\pi$
$281$	−1.63255e6	−1.23339	−0.616697	+	0.787201i	$0.711530\pi$
$282$	0	0
$283$	621056.	0.460962	0.230481	−	0.973077i	$-0.425970\pi$
$283$	621056.	0.460962	0.230481	+	0.973077i	$0.425970\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	181278.	0.129910
$288$	0	0
$289$	1.65907e6	1.16848
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−115908.	−0.0788759	−0.0394380	−	0.999222i	$-0.512557\pi$
$293$	−115908.	−0.0788759	−0.0394380	+	0.999222i	$0.512557\pi$
$294$	0	0
$295$	582063.	0.389417
$296$	0	0
$297$	0	0
$298$	0	0
$299$	30959.7	0.0200271
$300$	0	0
$301$	−828835.	−0.527293
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.41178e6	2.10006
$306$	0	0
$307$	765157.	0.463345	0.231673	−	0.972794i	$-0.425580\pi$
$307$	765157.	0.463345	0.231673	+	0.972794i	$0.425580\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	179636.	0.105315	0.0526576	−	0.998613i	$-0.483231\pi$
$311$	179636.	0.105315	0.0526576	+	0.998613i	$0.483231\pi$
$312$	0	0
$313$	895716.	0.516785	0.258392	−	0.966040i	$-0.416807\pi$
$313$	895716.	0.516785	0.258392	+	0.966040i	$0.416807\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−228258.	−0.127579	−0.0637893	−	0.997963i	$-0.520319\pi$
$317$	−228258.	−0.127579	−0.0637893	+	0.997963i	$0.520319\pi$
$318$	0	0
$319$	−2.40913e6	−1.32551
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.05852e6	0.564535
$324$	0	0
$325$	28945.7	0.0152011
$326$	0	0
$327$	0	0
$328$	0	0
$329$	32248.4	0.0164255
$330$	0	0
$331$	−3.01591e6	−1.51303	−0.756516	−	0.653975i	$-0.773100\pi$
$331$	−3.01591e6	−1.51303	−0.756516	+	0.653975i	$0.773100\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.23598e6	1.08857
$336$	0	0
$337$	−1.10209e6	−0.528619	−0.264310	−	0.964438i	$-0.585144\pi$
$337$	−1.10209e6	−0.528619	−0.264310	+	0.964438i	$0.585144\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.88323e6	2.73987
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.90191e6	−1.29378	−0.646891	−	0.762583i	$-0.723931\pi$
$347$	−2.90191e6	−1.29378	−0.646891	+	0.762583i	$0.723931\pi$
$348$	0	0
$349$	−2.23892e6	−0.983954	−0.491977	−	0.870608i	$-0.663726\pi$
$349$	−2.23892e6	−0.983954	−0.491977	+	0.870608i	$0.663726\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.05842e6	0.879217	0.439609	−	0.898189i	$-0.355117\pi$
$353$	2.05842e6	0.879217	0.439609	+	0.898189i	$0.355117\pi$
$354$	0	0
$355$	−721682.	−0.303931
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.96387e6	−0.804222	−0.402111	−	0.915591i	$-0.631723\pi$
$359$	−1.96387e6	−0.804222	−0.402111	+	0.915591i	$0.631723\pi$
$360$	0	0
$361$	−2.11219e6	−0.853031
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.32755e6	0.914464
$366$	0	0
$367$	−3.55132e6	−1.37634	−0.688169	−	0.725550i	$-0.741585\pi$
$367$	−3.55132e6	−1.37634	−0.688169	+	0.725550i	$0.741585\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.36667e6	0.515498
$372$	0	0
$373$	4.69562e6	1.74751	0.873757	−	0.486363i	$-0.161677\pi$
$373$	4.69562e6	1.74751	0.873757	+	0.486363i	$0.161677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−31468.5	−0.0114031
$378$	0	0
$379$	−957580.	−0.342434	−0.171217	−	0.985233i	$-0.554770\pi$
$379$	−957580.	−0.342434	−0.171217	+	0.985233i	$0.554770\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.79644e6	−0.625772	−0.312886	−	0.949791i	$-0.601296\pi$
$383$	−1.79644e6	−0.625772	−0.312886	+	0.949791i	$0.601296\pi$
$384$	0	0
$385$	2.85517e6	0.981702
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.36239e6	1.46167	0.730836	−	0.682553i	$-0.239130\pi$
$389$	4.36239e6	1.46167	0.730836	+	0.682553i	$0.239130\pi$
$390$	0	0
$391$	−5.57168e6	−1.84308
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.57048e6	−0.828937
$396$	0	0
$397$	3.84239e6	1.22356	0.611780	−	0.791028i	$-0.290454\pi$
$397$	3.84239e6	1.22356	0.611780	+	0.791028i	$0.290454\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.66668e6	1.13871	0.569354	−	0.822093i	$-0.307194\pi$
$401$	3.66668e6	1.13871	0.569354	+	0.822093i	$0.307194\pi$
$402$	0	0
$403$	76848.0	0.0235706
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.22801e6	2.76135
$408$	0	0
$409$	2.90345e6	0.858236	0.429118	−	0.903248i	$-0.358824\pi$
$409$	2.90345e6	0.858236	0.429118	+	0.903248i	$0.358824\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	365362.	0.105402
$414$	0	0
$415$	3.16344e6	0.901653
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.94202e6	1.09694	0.548470	−	0.836170i	$-0.315210\pi$
$419$	3.94202e6	1.09694	0.548470	+	0.836170i	$0.315210\pi$
$420$	0	0
$421$	−526504.	−0.144776	−0.0723880	−	0.997377i	$-0.523062\pi$
$421$	−526504.	−0.144776	−0.0723880	+	0.997377i	$0.523062\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.20923e6	−1.39895
$426$	0	0
$427$	2.14158e6	0.568414
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−387004.	−0.100351	−0.0501756	−	0.998740i	$-0.515978\pi$
$431$	−387004.	−0.100351	−0.0501756	+	0.998740i	$0.515978\pi$
$432$	0	0
$433$	−2.77123e6	−0.710318	−0.355159	−	0.934806i	$-0.615573\pi$
$433$	−2.77123e6	−0.710318	−0.355159	+	0.934806i	$0.615573\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.91551e6	−0.479822
$438$	0	0
$439$	3.01147e6	0.745791	0.372895	−	0.927873i	$-0.378365\pi$
$439$	3.01147e6	0.745791	0.372895	+	0.927873i	$0.378365\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.30598e6	−0.800370	−0.400185	−	0.916434i	$-0.631054\pi$
$443$	−3.30598e6	−0.800370	−0.400185	+	0.916434i	$0.631054\pi$
$444$	0	0
$445$	3.22799e6	0.772737
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.93077e6	−0.686066	−0.343033	−	0.939323i	$-0.611454\pi$
$449$	−2.93077e6	−0.686066	−0.343033	+	0.939323i	$0.611454\pi$
$450$	0	0
$451$	−2.76149e6	−0.639296
$452$	0	0
$453$	0	0
$454$	0	0
$455$	37294.8	0.00844539
$456$	0	0
$457$	−7.55328e6	−1.69178	−0.845892	−	0.533354i	$-0.820931\pi$
$457$	−7.55328e6	−1.69178	−0.845892	+	0.533354i	$0.820931\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.54017e6	−1.65245	−0.826226	−	0.563339i	$-0.809517\pi$
$461$	−7.54017e6	−1.65245	−0.826226	+	0.563339i	$0.809517\pi$
$462$	0	0
$463$	−2.69250e6	−0.583719	−0.291859	−	0.956461i	$-0.594274\pi$
$463$	−2.69250e6	−0.583719	−0.291859	+	0.956461i	$0.594274\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.28377e6	1.33330	0.666651	−	0.745370i	$-0.267727\pi$
$467$	6.28377e6	1.33330	0.666651	+	0.745370i	$0.267727\pi$
$468$	0	0
$469$	1.40353e6	0.294639
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.26260e7	2.59485
$474$	0	0
$475$	−1.79090e6	−0.364198
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.63273e6	0.723426	0.361713	−	0.932290i	$-0.382192\pi$
$479$	3.63273e6	0.723426	0.361713	+	0.932290i	$0.382192\pi$
$480$	0	0
$481$	120538.	0.0237554
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.64780e6	1.66936
$486$	0	0
$487$	626847.	0.119767	0.0598837	−	0.998205i	$-0.480927\pi$
$487$	626847.	0.119767	0.0598837	+	0.998205i	$0.480927\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.21381e6	−0.601612	−0.300806	−	0.953685i	$-0.597256\pi$
$491$	−3.21381e6	−0.601612	−0.300806	+	0.953685i	$0.597256\pi$
$492$	0	0
$493$	5.66325e6	1.04942
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−453002.	−0.0822638
$498$	0	0
$499$	−1.64079e6	−0.294987	−0.147494	−	0.989063i	$-0.547121\pi$
$499$	−1.64079e6	−0.294987	−0.147494	+	0.989063i	$0.547121\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.47878e6	1.14176	0.570878	−	0.821035i	$-0.306603\pi$
$503$	6.47878e6	1.14176	0.570878	+	0.821035i	$0.306603\pi$
$504$	0	0
$505$	6.32099e6	1.10295
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.55190e6	−0.949833	−0.474916	−	0.880031i	$-0.657522\pi$
$509$	−5.55190e6	−0.949833	−0.474916	+	0.880031i	$0.657522\pi$
$510$	0	0
$511$	1.46101e6	0.247514
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.18893e6	−1.36053
$516$	0	0
$517$	−491253.	−0.0808311
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.57348e6	−1.22237	−0.611183	−	0.791490i	$-0.709306\pi$
$521$	−7.57348e6	−1.22237	−0.611183	+	0.791490i	$0.709306\pi$
$522$	0	0
$523$	6.16189e6	0.985053	0.492526	−	0.870298i	$-0.336074\pi$
$523$	6.16189e6	0.985053	0.492526	+	0.870298i	$0.336074\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.38300e7	−2.16918
$528$	0	0
$529$	3.64626e6	0.566512
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−36071.2	−0.00549974
$534$	0	0
$535$	1.51346e7	2.28606
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.79220e6	0.265713
$540$	0	0
$541$	−9.51400e6	−1.39756	−0.698779	−	0.715338i	$-0.746273\pi$
$541$	−9.51400e6	−1.39756	−0.698779	+	0.715338i	$0.746273\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.20957e7	1.74437
$546$	0	0
$547$	590098.	0.0843249	0.0421624	−	0.999111i	$-0.486575\pi$
$547$	590098.	0.0843249	0.0421624	+	0.999111i	$0.486575\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.94699e6	0.273202
$552$	0	0
$553$	−1.61350e6	−0.224365
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.71272e6	−0.507054	−0.253527	−	0.967328i	$-0.581591\pi$
$557$	−3.71272e6	−0.507054	−0.253527	+	0.967328i	$0.581591\pi$
$558$	0	0
$559$	164923.	0.0223230
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.24074e6	−1.22867	−0.614336	−	0.789045i	$-0.710576\pi$
$563$	−9.24074e6	−1.22867	−0.614336	+	0.789045i	$0.710576\pi$
$564$	0	0
$565$	7.14884e6	0.942138
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.73509e6	−0.872093	−0.436047	−	0.899924i	$-0.643622\pi$
$569$	−6.73509e6	−0.872093	−0.436047	+	0.899924i	$0.643622\pi$
$570$	0	0
$571$	3.66513e6	0.470435	0.235217	−	0.971943i	$-0.424420\pi$
$571$	3.66513e6	0.470435	0.235217	+	0.971943i	$0.424420\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.42670e6	1.18902
$576$	0	0
$577$	−8.73504e6	−1.09226	−0.546129	−	0.837701i	$-0.683899\pi$
$577$	−8.73504e6	−1.09226	−0.546129	+	0.837701i	$0.683899\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.98570e6	0.244047
$582$	0	0
$583$	−2.08190e7	−2.53681
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.06478e6	0.966045	0.483022	−	0.875608i	$-0.339539\pi$
$587$	8.06478e6	0.966045	0.483022	+	0.875608i	$0.339539\pi$
$588$	0	0
$589$	−4.75467e6	−0.564718
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.19273e6	−0.723178	−0.361589	−	0.932338i	$-0.617766\pi$
$593$	−6.19273e6	−0.723178	−0.361589	+	0.932338i	$0.617766\pi$
$594$	0	0
$595$	−6.71179e6	−0.777223
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.30764e6	−0.262786	−0.131393	−	0.991330i	$-0.541945\pi$
$599$	−2.30764e6	−0.262786	−0.131393	+	0.991330i	$0.541945\pi$
$600$	0	0
$601$	−1.09721e7	−1.23909	−0.619545	−	0.784961i	$-0.712683\pi$
$601$	−1.09721e7	−1.23909	−0.619545	+	0.784961i	$0.712683\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.09219e7	−3.43461
$606$	0	0
$607$	−1.32918e7	−1.46424	−0.732121	−	0.681175i	$-0.761469\pi$
$607$	−1.32918e7	−1.46424	−0.732121	+	0.681175i	$0.761469\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6416.85	−0.000695374 0
$612$	0	0
$613$	−1.17502e7	−1.26298	−0.631488	−	0.775385i	$-0.717556\pi$
$613$	−1.17502e7	−1.26298	−0.631488	+	0.775385i	$0.717556\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.26344e6	−0.239362	−0.119681	−	0.992812i	$-0.538187\pi$
$617$	−2.26344e6	−0.239362	−0.119681	+	0.992812i	$0.538187\pi$
$618$	0	0
$619$	1.23588e7	1.29643	0.648215	−	0.761457i	$-0.275516\pi$
$619$	1.23588e7	1.29643	0.648215	+	0.761457i	$0.275516\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.02621e6	0.209154
$624$	0	0
$625$	−1.02295e7	−1.04750
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.16928e7	−2.18619
$630$	0	0
$631$	−2.74069e6	−0.274023	−0.137012	−	0.990569i	$-0.543750\pi$
$631$	−2.74069e6	−0.274023	−0.137012	+	0.990569i	$0.543750\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.60513e6	−0.846883
$636$	0	0
$637$	23410.0	0.00228588
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.49344e7	1.43563	0.717816	−	0.696232i	$-0.245142\pi$
$641$	1.49344e7	1.43563	0.717816	+	0.696232i	$0.245142\pi$
$642$	0	0
$643$	1.12339e7	1.07153	0.535764	−	0.844368i	$-0.320024\pi$
$643$	1.12339e7	1.07153	0.535764	+	0.844368i	$0.320024\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.29688e6	0.497461	0.248731	−	0.968573i	$-0.419987\pi$
$647$	5.29688e6	0.497461	0.248731	+	0.968573i	$0.419987\pi$
$648$	0	0
$649$	−5.56572e6	−0.518692
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.08138e7	−0.992418	−0.496209	−	0.868203i	$-0.665275\pi$
$653$	−1.08138e7	−0.992418	−0.496209	+	0.868203i	$0.665275\pi$
$654$	0	0
$655$	2.82091e7	2.56913
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−417354.	−0.0374362	−0.0187181	−	0.999825i	$-0.505958\pi$
$659$	−417354.	−0.0374362	−0.0187181	+	0.999825i	$0.505958\pi$
$660$	0	0
$661$	6.80352e6	0.605661	0.302831	−	0.953044i	$-0.402068\pi$
$661$	6.80352e6	0.605661	0.302831	+	0.953044i	$0.402068\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.30747e6	−0.202340
$666$	0	0
$667$	−1.02483e7	−0.891944
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.26236e7	−2.79721
$672$	0	0
$673$	−1.43365e7	−1.22013	−0.610066	−	0.792351i	$-0.708857\pi$
$673$	−1.43365e7	−1.22013	−0.610066	+	0.792351i	$0.708857\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.09804e6	−0.343641	−0.171820	−	0.985128i	$-0.554965\pi$
$677$	−4.09804e6	−0.343641	−0.171820	+	0.985128i	$0.554965\pi$
$678$	0	0
$679$	5.42824e6	0.451840
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.16379e6	0.669638	0.334819	−	0.942282i	$-0.391325\pi$
$683$	8.16379e6	0.669638	0.334819	+	0.942282i	$0.391325\pi$
$684$	0	0
$685$	2.29225e7	1.86654
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−271942.	−0.0218237
$690$	0	0
$691$	9.40335e6	0.749182	0.374591	−	0.927190i	$-0.377783\pi$
$691$	9.40335e6	0.749182	0.374591	+	0.927190i	$0.377783\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.22401e6	−0.567305
$696$	0	0
$697$	6.49157e6	0.506137
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.76733e7	1.35838	0.679192	−	0.733961i	$-0.262330\pi$
$701$	1.76733e7	1.35838	0.679192	+	0.733961i	$0.262330\pi$
$702$	0	0
$703$	−7.45783e6	−0.569147
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.96770e6	0.298531
$708$	0	0
$709$	1.61349e7	1.20546	0.602728	−	0.797947i	$-0.294080\pi$
$709$	1.61349e7	1.20546	0.602728	+	0.797947i	$0.294080\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.50270e7	1.84368
$714$	0	0
$715$	−568127.	−0.0415605
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.10160e7	−1.51610	−0.758051	−	0.652195i	$-0.773848\pi$
$719$	−2.10160e7	−1.51610	−0.758051	+	0.652195i	$0.773848\pi$
$720$	0	0
$721$	−5.14021e6	−0.368250
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.58163e6	−0.677009
$726$	0	0
$727$	2.51729e7	1.76643	0.883217	−	0.468964i	$-0.155373\pi$
$727$	2.51729e7	1.76643	0.883217	+	0.468964i	$0.155373\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.96805e7	−2.05437
$732$	0	0
$733$	−1.80996e7	−1.24425	−0.622127	−	0.782916i	$-0.713731\pi$
$733$	−1.80996e7	−1.24425	−0.622127	+	0.782916i	$0.713731\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.13806e7	−1.44994
$738$	0	0
$739$	−9.69580e6	−0.653089	−0.326545	−	0.945182i	$-0.605884\pi$
$739$	−9.69580e6	−0.653089	−0.326545	+	0.945182i	$0.605884\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.72447e6	−0.313965	−0.156982	−	0.987601i	$-0.550177\pi$
$743$	−4.72447e6	−0.313965	−0.156982	+	0.987601i	$0.550177\pi$
$744$	0	0
$745$	1.02878e6	0.0679099
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.50004e6	0.618758
$750$	0	0
$751$	3.00055e7	1.94134	0.970668	−	0.240422i	$-0.0772859\pi$
$751$	3.00055e7	1.94134	0.970668	+	0.240422i	$0.0772859\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.37989e7	1.51946
$756$	0	0
$757$	907379.	0.0575505	0.0287752	−	0.999586i	$-0.490839\pi$
$757$	907379.	0.0575505	0.0287752	+	0.999586i	$0.490839\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.22057e7	0.764014	0.382007	−	0.924159i	$-0.375233\pi$
$761$	1.22057e7	0.764014	0.382007	+	0.924159i	$0.375233\pi$
$762$	0	0
$763$	7.59248e6	0.472141
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−72700.6	−0.00446220
$768$	0	0
$769$	−2.85670e7	−1.74200	−0.871000	−	0.491283i	$-0.836528\pi$
$769$	−2.85670e7	−1.74200	−0.871000	+	0.491283i	$0.836528\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−905880.	−0.0545283	−0.0272642	−	0.999628i	$-0.508680\pi$
$773$	−905880.	−0.0545283	−0.0272642	+	0.999628i	$0.508680\pi$
$774$	0	0
$775$	2.33989e7	1.39940
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.23176e6	0.131766
$780$	0	0
$781$	6.90076e6	0.404827
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.99992e6	0.289593
$786$	0	0
$787$	−1.25311e7	−0.721192	−0.360596	−	0.932722i	$-0.617427\pi$
$787$	−1.25311e7	−0.721192	−0.360596	+	0.932722i	$0.617427\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.48735e6	0.255005
$792$	0	0
$793$	−426136.	−0.0240639
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.14513e6	−0.175385	−0.0876924	−	0.996148i	$-0.527949\pi$
$797$	−3.14513e6	−0.175385	−0.0876924	+	0.996148i	$0.527949\pi$
$798$	0	0
$799$	1.15481e6	0.0639947
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.22561e7	−1.21804
$804$	0	0
$805$	1.21458e7	0.660595
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.26907e6	0.175611	0.0878057	−	0.996138i	$-0.472015\pi$
$809$	3.26907e6	0.175611	0.0878057	+	0.996138i	$0.472015\pi$
$810$	0	0
$811$	566320.	0.0302350	0.0151175	−	0.999886i	$-0.495188\pi$
$811$	566320.	0.0302350	0.0151175	+	0.999886i	$0.495188\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.81817e7	−2.54090
$816$	0	0
$817$	−1.02040e7	−0.534828
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.98337e6	−0.102694	−0.0513471	−	0.998681i	$-0.516351\pi$
$821$	−1.98337e6	−0.102694	−0.0513471	+	0.998681i	$0.516351\pi$
$822$	0	0
$823$	−3.14336e7	−1.61769	−0.808844	−	0.588024i	$-0.799906\pi$
$823$	−3.14336e7	−1.61769	−0.808844	+	0.588024i	$0.799906\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.44099e7	0.732651	0.366325	−	0.930487i	$-0.380616\pi$
$827$	1.44099e7	0.732651	0.366325	+	0.930487i	$0.380616\pi$
$828$	0	0
$829$	−3.60154e7	−1.82013	−0.910063	−	0.414469i	$-0.863967\pi$
$829$	−3.60154e7	−1.82013	−0.910063	+	0.414469i	$0.863967\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.21300e6	−0.210368
$834$	0	0
$835$	3.86170e7	1.91673
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.05418e7	1.00747	0.503737	−	0.863857i	$-0.331958\pi$
$839$	2.05418e7	1.00747	0.503737	+	0.863857i	$0.331958\pi$
$840$	0	0
$841$	−1.00944e7	−0.492142
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.89766e7	1.39607
$846$	0	0
$847$	−1.94098e7	−0.929633
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.92556e7	1.85814
$852$	0	0
$853$	−3.21606e7	−1.51339	−0.756695	−	0.653768i	$-0.773187\pi$
$853$	−3.21606e7	−1.51339	−0.756695	+	0.653768i	$0.773187\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.87973e7	−1.33937	−0.669683	−	0.742647i	$-0.733570\pi$
$857$	−2.87973e7	−1.33937	−0.669683	+	0.742647i	$0.733570\pi$
$858$	0	0
$859$	−2.72233e7	−1.25880	−0.629401	−	0.777081i	$-0.716699\pi$
$859$	−2.72233e7	−1.25880	−0.629401	+	0.777081i	$0.716699\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	56328.3	0.00257454	0.00128727	−	0.999999i	$-0.499590\pi$
$863$	56328.3	0.00257454	0.00128727	+	0.999999i	$0.499590\pi$
$864$	0	0
$865$	−5.47576e7	−2.48831
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.45791e7	1.10412
$870$	0	0
$871$	−279277.	−0.0124736
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−597669.	−0.0263901
$876$	0	0
$877$	2.32650e7	1.02142	0.510710	−	0.859753i	$-0.329383\pi$
$877$	2.32650e7	1.02142	0.510710	+	0.859753i	$0.329383\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.92252e7	1.70265	0.851326	−	0.524638i	$-0.175799\pi$
$881$	3.92252e7	1.70265	0.851326	+	0.524638i	$0.175799\pi$
$882$	0	0
$883$	2.86152e7	1.23508	0.617540	−	0.786540i	$-0.288130\pi$
$883$	2.86152e7	1.23508	0.617540	+	0.786540i	$0.288130\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.81794e6	0.248290	0.124145	−	0.992264i	$-0.460381\pi$
$887$	5.81794e6	0.248290	0.124145	+	0.992264i	$0.460381\pi$
$888$	0	0
$889$	−5.40146e6	−0.229222
$890$	0	0
$891$	0	0
$892$	0	0
$893$	397017.	0.0166602
$894$	0	0
$895$	1.54134e7	0.643191
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.54383e7	−1.04976
$900$	0	0
$901$	4.89402e7	2.00842
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.78606e7	−2.75420
$906$	0	0
$907$	9.55467e6	0.385654	0.192827	−	0.981233i	$-0.438234\pi$
$907$	9.55467e6	0.385654	0.192827	+	0.981233i	$0.438234\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.36257e6	0.214081	0.107040	−	0.994255i	$-0.465863\pi$
$911$	5.36257e6	0.214081	0.107040	+	0.994255i	$0.465863\pi$
$912$	0	0
$913$	−3.02490e7	−1.20097
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.77069e7	0.695375
$918$	0	0
$919$	1.97667e6	0.0772050	0.0386025	−	0.999255i	$-0.487709\pi$
$919$	1.97667e6	0.0772050	0.0386025	+	0.999255i	$0.487709\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	90139.2	0.00348265
$924$	0	0
$925$	3.67019e7	1.41037
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.09537e7	−0.416412	−0.208206	−	0.978085i	$-0.566762\pi$
$929$	−1.09537e7	−0.416412	−0.208206	+	0.978085i	$0.566762\pi$
$930$	0	0
$931$	−1.44840e6	−0.0547666
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.02243e8	3.82478
$936$	0	0
$937$	−6.80229e6	−0.253108	−0.126554	−	0.991960i	$-0.540392\pi$
$937$	−6.80229e6	−0.253108	−0.126554	+	0.991960i	$0.540392\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.00214e6	−0.110524	−0.0552620	−	0.998472i	$-0.517599\pi$
$941$	−3.00214e6	−0.110524	−0.0552620	+	0.998472i	$0.517599\pi$
$942$	0	0
$943$	−1.17473e7	−0.430187
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.85814e7	1.76033	0.880166	−	0.474665i	$-0.157431\pi$
$947$	4.85814e7	1.76033	0.880166	+	0.474665i	$0.157431\pi$
$948$	0	0
$949$	−290714.	−0.0104785
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.24058e7	1.15582	0.577911	−	0.816100i	$-0.303868\pi$
$953$	3.24058e7	1.15582	0.577911	+	0.816100i	$0.303868\pi$
$954$	0	0
$955$	9.33036e6	0.331047
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.43885e7	0.505208
$960$	0	0
$961$	3.34928e7	1.16988
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.33421e7	−0.461218
$966$	0	0
$967$	2.37554e7	0.816950	0.408475	−	0.912769i	$-0.366061\pi$
$967$	2.37554e7	0.816950	0.408475	+	0.912769i	$0.366061\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.27649e7	−1.11522	−0.557610	−	0.830103i	$-0.688282\pi$
$971$	−3.27649e7	−1.11522	−0.557610	+	0.830103i	$0.688282\pi$
$972$	0	0
$973$	−4.53453e6	−0.153550
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.09726e7	−1.70844	−0.854222	−	0.519908i	$-0.825966\pi$
$977$	−5.09726e7	−1.70844	−0.854222	+	0.519908i	$0.825966\pi$
$978$	0	0
$979$	−3.08662e7	−1.02926
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.63735e7	1.20061	0.600304	−	0.799772i	$-0.295046\pi$
$983$	3.63735e7	1.20061	0.600304	+	0.799772i	$0.295046\pi$
$984$	0	0
$985$	−5.58217e7	−1.83321
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.37104e7	1.74609
$990$	0	0
$991$	7.46245e6	0.241378	0.120689	−	0.992690i	$-0.461490\pi$
$991$	7.46245e6	0.241378	0.120689	+	0.992690i	$0.461490\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.58699e7	1.78904
$996$	0	0
$997$	−373648.	−0.0119049	−0.00595243	−	0.999982i	$-0.501895\pi$
$997$	−373648.	−0.0119049	−0.00595243	+	0.999982i	$0.501895\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.a.n.1.1	✓	2
3.2	odd	2		504.6.a.q.1.2	yes	2
4.3	odd	2		1008.6.a.bj.1.1		2
12.11	even	2		1008.6.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.6.a.n.1.1	✓	2	1.1	even	1	trivial
504.6.a.q.1.2	yes	2	3.2	odd	2
1008.6.a.bj.1.1		2	4.3	odd	2
1008.6.a.br.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)

\(f(q)\)	\(=\)	\(q-78.0625 q^{5} -49.0000 q^{7} +746.437 q^{11} +9.75012 q^{13} -1754.69 q^{17} -603.250 q^{19} +3175.31 q^{23} +2968.75 q^{25} -3227.50 q^{29} +7881.75 q^{31} +3825.06 q^{35} +12362.7 q^{37} -3699.56 q^{41} +16915.0 q^{43} -658.130 q^{47} +2401.00 q^{49} -27891.1 q^{53} -58268.7 q^{55} -7456.38 q^{59} -43705.7 q^{61} -761.119 q^{65} -28643.5 q^{67} +9244.93 q^{71} -29816.5 q^{73} -36575.4 q^{77} +32928.5 q^{79} -40524.5 q^{83} +136975. q^{85} -41351.3 q^{89} -477.756 q^{91} +47091.2 q^{95} -110780. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 28 q^{5} - 98 q^{7} + 596 q^{11} + 532 q^{13} - 2100 q^{17} - 2744 q^{19} + 3660 q^{23} + 2350 q^{25} + 720 q^{29} + 3976 q^{31} + 1372 q^{35} + 6788 q^{37} + 4004 q^{41} + 19480 q^{43} - 21560 q^{47}+ \cdots - 36036 q^{97}+O(q^{100}) \) 2 * q - 28 * q^5 - 98 * q^7 + 596 * q^11 + 532 * q^13 - 2100 * q^17 - 2744 * q^19 + 3660 * q^23 + 2350 * q^25 + 720 * q^29 + 3976 * q^31 + 1372 * q^35 + 6788 * q^37 + 4004 * q^41 + 19480 * q^43 - 21560 * q^47 + 4802 * q^49 - 57576 * q^53 - 65800 * q^55 - 22344 * q^59 - 38724 * q^61 + 25384 * q^65 + 7288 * q^67 - 3932 * q^71 + 19292 * q^73 - 29204 * q^77 + 15632 * q^79 - 22624 * q^83 + 119688 * q^85 - 112812 * q^89 - 26068 * q^91 - 60080 * q^95 - 36036 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more